Complex Numbers: Defining an Ordered System

[lostcauses10x]

Complex numbers?
Since the system is not an ordered pair, how then is it defined using the complex system as an ordered system to plot the z axis (Plane) to use a function?
At the point we input each point of the Real and imaginary plane into a function to get out an answer in the Z plane, is it now an ordered pair per point on input ?

 

[DrClaude]

A complex number is an ordered pair.

 

[SteamKing]

In complex analysis, the relation w = f(z), where f is some function and w and z are complex number, is thought of as a mapping, whereby for each number z, the function f(z) points to, or 'maps', a different complex number w in a different complex plane.

 

[pwsnafu]

A complex number is an ordered pair.

To be precise, we construct the complex numbers from a ordered pairs of real numbers by defining addition and multiplication.

 

[lostcauses10x]

pwsnafu
Where as I believe it is correct, in the link you gave: it is a statement given without any reference or source.

 

[Number Nine]

pwsnafu
Where as I believe it is correct, in the link you gave: it is a statement given without any reference or source.

It's the standard construction of the complex numbers. If you want a reference, see any text on complex analysis of algebra ever published.

 

[lostcauses10x]

Hey folks thanks. Simply put based of the limited ability to define i to the real set we created an set with the real and imaginary that is an analogy. of course from there end up with complex analysis etc.
Not sure what the "Mon" replies are, seem to add noting to the discussion.